Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{q^3 - 5q^2 - 24q}{-2q^3 + 2q^2 + 112q}$
First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {q(q^2 - 5q - 24)} {-2q(q^2 - q - 56)} $ $ n = -\dfrac{q}{2q} \cdot \dfrac{q^2 - 5q - 24}{q^2 - q - 56} $ Simplify: $ n = - \dfrac{1}{2} \cdot \dfrac{q^2 - 5q - 24}{q^2 - q - 56}$ Since we are dividing by $q$ , we must remember that $q \neq 0$ Next factor the numerator and denominator. $ n = - \dfrac{1}{2} \cdot \dfrac{(q - 8)(q + 3)}{(q - 8)(q + 7)}$ Assuming $q \neq 8$ , we can cancel the $q - 8$ $ n = - \dfrac{1}{2} \cdot \dfrac{q + 3}{q + 7}$ Therefore: $ n = \dfrac{ -q - 3 }{ 2(q + 7)}$, $q \neq 8$, $q \neq 0$